428 ELECTRICAL MEASUREMENTS 



tension, TI, has been applied to the springs by means of the 

 micrometer screw. 



When the angular velocity is o> and the center of gravity is r 

 cm. from the axis, the centrifugal force will be 



K c = McoV 

 and the tension on the springs will be 



K s = C(r - a') + Ti 

 At equilibrium K c = K s and 



= C(r - a'} + TV 



The equilibrium may be stable, neutral, or unstable, depending 

 on the initial tension T\. For, suppose that at some instant 

 when the center of gravity of the weight is distant r from the 

 axis, K c happens to be equal to K s . If the weight is given a 

 slight displacement outward, 5r, 



8K C 8r 5K S Cdr 8r 



= 7 and ^ = 



r 



^r is the initial extension of the spring. It will be denoted by 



e'\ then 



dK s dr 



K 8 r - (a' - e') 



If a' is greater than e' the denominator of the expression for 

 &K bK &K 



-jr~ is less than r, -~-^ > -j^- and the weight will return toward 

 AS AS A c 



its original position, that is, the equilibrium is stable. If 

 a' = e', then -r^ = -^ and the equilibrium is neutral. If 



AS AC 



*TT *J 



a' is less than e' the denominator is greater than r and -^- <-^ 



In this case, the weight will suddenly fly outward to the extent 

 of its travel, the spring being insufficient to make it return to its 

 original position. This is the case of unstable equilibrium. 



If the springs be given increasing tension the equilibrium 

 remains stable until T\ = Co,' or e' = a', that is, until the tension 

 is that which would bring the center of gravity of the weight to 



